The present invention relates generally to machine learning, and more particularly to support vector machines.
Machine learning involves techniques to allow computers to “learn”. More specifically, machine learning involves training a computer system to perform some task, rather than directly programming the system to perform the task. The system observes some data and automatically determines some structure of the data for use at a later time when processing unknown data.
Machine learning techniques generally create a function from training data. The training data consists of pairs of input objects (typically vectors), and desired outputs. The output of the function can be a continuous value (called regression), or can predict a class label of the input object (called classification). The task of the learning machine is to predict the value of the function for any valid input object after having seen only a small number of training examples (i.e. pairs of input and target output).
One particular type of learning machine is a support vector machine (SVM). SVMs are well known in the art, for example as described in V. Vapnik, Statistical Learning Theory, Wiley, New York, 1998; and C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery 2, 121-167, 1998. Although well known, a brief description of SVMs will be given here in order to aid in the following description of the present invention.
Consider the classification shown in FIG. 1 which shows data having the classification of circle or square. The question becomes, what is the best way of dividing the two classes? An SVM creates a maximum-margin hyperplane defined by support vectors as shown in FIG. 2. The support vectors are shown as 202, 204 and 206 and they define those input vectors of the training data which are used as classification boundaries to define the hyperplane 208. The goal in defining a hyperplane in a classification problem is to maximize the margin (w) 210 which is the distance between the support vectors of each different class. In other words, the maximum-margin hyperplane splits the training examples such that the distance from the closest support vectors is maximized. The support vectors are determined by solving a quadratic programming (QP) optimization problem. There exist several well known QP algorithms for use with SVMs, for example as described in R. Fletcher, Practical Methods of Optimization, Wiley, New York, 2001; and M. S. Bazaraa, H. D. Shrali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley Interscience, New York, 1993. Only a small subset of the training data vectors (i.e., the support vectors) need to be considered in order to determine the optimal hyperplane. Thus, the problem of defining the support vectors may also be considered a filtering problem. More particularly, the job of the SVM during the training phase is to filter out the training data vectors which are not support vectors.
As can be seen from FIG. 2, the optimal hyperplane 208 is linear, which assumes that the data to be classified is linearly separable. However, this is not always the case. For example, consider FIG. 3 in which the data is classified into two sets (X and O). As shown on the left side of the figure, in one dimensional space the two classes are not linearly separable. However, by mapping the one dimensional data into 2 dimensional space as shown on the right side of the figure, the data becomes linearly separable by line 302. This same idea is shown in FIG. 4, which, on the left side of the figure, shows two dimensional data with the classification boundaries defined by support vectors (shown as disks with outlines around them). However, the class divider 402 is a curve, not a line, and the two dimensional data are not linearly separable. However, by mapping the two dimensional data into higher dimensional space as shown on the right side of FIG. 4, the data becomes linearly separable by hyperplane 404. The mapping function that calculates dot products between vectors in the space of higher dimensionality is called a kernel and is generally referred to herein as k. The use of the kernel function to map data from a lower to a higher dimensionality is well known in the art, for example as described in V. Vapnik, Statistical Learning Theory, Wiley, New York, 1998.
After the SVM is trained as described above, input data may be classified by applying the following equation:
  y  =      sign    ⁡          (                                    ∑                          i              =              1                        M                    ⁢                                          ⁢                                    α              i                        ⁢                          k              ⁡                              (                                                      x                    i                                    ,                  x                                )                                                    -        b            )      where xi represents the support vectors, x is the vector to be classified, αi and b are parameters obtained by the training algorithm, and y is the class label that is assigned to the vector being classified.
The equation k(x,xi)=exp(−∥x−xi∥2/c) is an example of a kernel function, namely a radial basis function. Other types of kernel functions may be used as well.
Although SVMs are powerful classification and regression tools, one disadvantage is that their computation and storage requirements increase rapidly with the number of training vectors, putting many problems of practical interest out of their reach. As described above, the core of an SVM is a quadratic programming problem, separating support vectors from the rest of the training data. General-purpose QP solvers tend to scale with the cube of the number of training vectors (O(k3)). Specialized algorithms, typically based on gradient descent methods, achieve gains in efficiency, but still become impractically slow for problem sizes on the order of 100,000 training vectors (2-class problems).
One existing approach for accelerating the QP is based on ‘chunking’ where subsets of the training data are optimized iteratively, until the global optimum is reached. This technique is described in B. Boser, I. Guyon. V. Vapnik, “A training algorithm for optimal margin classifiers” in Proc. 5th Annual Workshop on Computational Learning Theory, Pittsburgh, ACM, 1992; E. Osuna, R. Freund, F. Girosi, “Training Support Vector Machines, an Application to Face Detection”, in Computer vision and Pattem Recognition, pp. 130-136, 1997; and T. Joachims, “Making large-scale support vector machine learning practical”, in Advances in Kernel Methods, B. Schölkopf, C. Burges, A. Smola, (eds.), Cambridge, MIT Press, 1998. ‘Sequential Minimal Optimization’ (SMO), as described in J. C. Platt, “Fast training of support vector machines using sequential minimal optimization”, in Adv. in Kernel Methods: Schölkopf, C. Burges, A. Simola (eds.), 1998 reduces the chunk size to 2 vectors, and is the most popular of these chunking algorithms. Eliminating non-support vectors early during the optimization process is another strategy that provides substantial savings in computation. Efficient SVM implementations incorporate steps known as ‘shrinking’ for early identification of non-support vectors, as described in T. Joachims, “Making large-scale support vector machine learning practical”, in Advances in Kernel Methods, B. Schölkopf, C. Burges, A. Smola, (eds.), Cambridge, MIT Press, 1998; and R. Collobert, S. Bengio, and J. Mariéthoz, “Torch: A modular machine learning software library”, Technical Report IDIAP-RR 02-46, IDIAP, 2002. In combination with caching of the kernel data, these techniques reduce the computation requirements by orders of magnitude. Another approach, named ‘digesting’, and described in D. DeCoste and B. Schölkopf, “Training Invariant Support Vector Machines”, Machine Learning 46, 161-190, 2002 optimizes subsets closer to completion before adding new data, thereby saving considerable amounts of storage.
Improving SVM compute-speed through parallelization is difficult due to dependencies between the computation steps. Parallelizations have been attempted by splitting the problem into smaller subsets that can be optimized independently, either through initial clustering of the data or through a trained combination of the results from individually optimized subsets as described in R. Collobert, Y. Bengio, S. Bengio, “A Parallel Mixture of SVMs for Very Large Scale Problems”, in Neural Information Processing Systems, Vol. 17, MIT Press, 2004. If a problem can be structured in this way, data-parallelization can be efficient. However, for many problems, it is questionable whether, after splitting into smaller problems, a global optimum can be found. Variations of the standard SVM algorithm, such as the Proximal SVM as described in A. Tveit, H. Engum, “Parallelization of the Incremental Proximal Support Vector Machine Classifier using a Heap-based Tree Topology”, Tech. Report, IDI, NTNU, Trondheim, 2003 are better suited for parallelization, but their performance and applicability to high-dimensional problems remain questionable. Another parallelization scheme as described in J. X. Dong, A. Krzyzak, C. Y. Suen, “A fast Parallel Optimization for Training Support Vector Machine.” Proceedings of 3rd International Conference on Machine Learning and Data Mining, P. Perner and A. Rosenfeld (Eds.), Springer Lecture Notes in Artificial Intelligence (LNAI 2734), pp. 96-105, Leipzig, Germany, Jul. 5-7, 2003, approximates the kernel matrix by a block-diagonal.
Other parallelization techniques also exist. G. Zanghirati, L. Zanni, “A Parallel Solver for Large Quadratic Programs in Training Support Vector Machines”, Parallel Computing, 29, 535-551, 2003, describes a parallel optimization algorithm based on gradient projections that uses a spectral gradient method for fast convergence. U.S. patent application Ser. No. 10/978,129, filed Oct. 29, 2004, entitled Parallel Support Vector Method and Apparatus, describes a parallelization technique that breaks a training data set into smaller subsets that are optimized individually in a distributed manner. Interior point (IP) methods are known for solving large quadratic programming (QP) problems, and parallelizations have been proposed, for example in M. D'Appuzo, M. Marino, “Parallel Computational Issues of an Interior Point Method for Solving Large Bound-Constrained Quadratic Programming Problems”, Parallel Computing, 29, 467-483, 2003.
Although SVMs, and the above described techniques, are powerful regression and classification tools, they do not scale well to very large problems.